Find the coefficient of $x^{15}$

Question

How do you find the coefficient of $x^{15}$ from $x^{3}$$(1-2x)^{10}$? Thank you.

Answer 1

The coefficient of $x^{15}$ from $\displaystyle x^3(1−2x)^{10}$

$=$ the coefficient of $\displaystyle x^{15-3}$ from $\displaystyle(1−2x)^{10}$

Now the $r$th term of $\displaystyle(1−2x)^{10}$ is $\displaystyle\binom{10}r(1)^{10-r}(-2x)^r$

Clearly, we need $r=12$

Now, $\displaystyle\binom{10}{12}=?$